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Best of Donald Catlin

Gaming Guru

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Flush Meandering

15 April 2005

I'm sure we all make mistakes playing Video Poker; I know I do. Not too long ago I was playing 9/6 Jacks and was dealt 2H, AC, KD, JD, 2S. Well that King and Jack two-card Royal leapt right off the screen at me and before I knew it I was drawing three to it. Wrong! Now I do a lot of stupid things but for once this wasn't stupidity. I know the proper play; keep the low pair. My problem there was lack of concentration. My mind was somewhere else. I think a lot of players suffer from this malady. It might be that you're thinking about an appointment, or what you're going to have for dinner, or a really sexy cocktail waitress who just went by (although there seems to be less of them nowadays than in old Vegas), or maybe you're just thinking about the strange way the game sometimes unfolds. Thinking about the game and concentrating on the game are really two completely different things though. This brings me to the following mind meandering story.

In January I was playing 9/6 Jacks on a Triple Play machine at the Golden Nugget in downtown Las Vegas. There is a bank of machines that offer this game in the hallway leading south out of the casino to the hotel rooms. In a very few minutes I had been dealt several four-card Flushes and had not filled a single one of them. Drat! Then I was dealt a pat five-card Flush. I got to thinking (a bad move) that I seem to fill four-card Flushes with less frequency than I think I should and I get dealt pat Flushes more frequently than seems reasonable to me. Of course, I don't keep score -- I'm just talking about a feeling, an emotion, intuition, or some other unreliable synapse that was fired up by my poor old brain. Oh yeah, at this point I also made another playing error.

I decided to take a look at this issue at my leisure, a much better plan, by the way, than musing about it while playing. Specifically, I decided to address this question. How frequently does one get dealt a pat Flush and play the Flush (this isn't as strange as it sounds) and how often does one get dealt a four Flush and successfully draw and get a five Flush? Immediately you can see that there are problems. If the five Flush contains a four-card Royal, then the low card should be discarded and the player should draw one (I discussed this in an earlier article). So does this situation count as a five Flush or not? If one draws a four Flush that contains a three-card Royal, then the low suited card should be discarded and the player should draw two. Does this count as drawing on a four Flush? Well, you get the idea; we have to make some choices.

Now understand I'm not trying to calculate expected returns here or the chances of winning big (or small); I'm simply addressing the above question with an eye on the emotional issue of the player's perception. Thus if the player gets dealt a pat five Flush I will count this as a five Flush result only if the player's final hand is a five Flush. If the player gets dealt a four Flush that is also a straight I won't count this since the player will not draw to the four suited cards. Same decision if the player has a high pair (unless, of course he has a four-card Straight Flush which takes precedence over the high pair). If the player has a four Flush that contains a three-card Royal, then I'll count it if the player draws two to the Royal and obtains a Flush. Look at the original question.

Very well, using the stuff from my January article it is easy to figure the number of ways to draw a pat Flush. There are four ways to choose the suit and once this is done there are C(13, 5) ways to select five of the thirteen suited cards. Thus

# Pat Flushes = 4 x C(13, 5) = 5148 (1)

Note that this number includes Royal and Straight Flushes.

For the four Flushes there are again four ways to choose the suit, then C(13, 4) ways to choose four of the suited cards, and finally 39 ways to choose the remaining off-suit card. Thus

# Four Flushes = 4 x C913, 4) x 39 = 111,540 (2)

It turns out that if you want to do the necessary calculations by hand it is easier if you subdivide the four Flushes into hands containing zero through four high cards in the Flush. Since there are four ways to choose the suit of the four and then three ways to choose the off-suit, we can count things up using a specific situation (say four Clubs and a Diamond) and then multiply the result by 12. Let me show you an example.

Suppose that I want to calculate the number of five-card Straights in a four flush with no high Flush cards. I simply list them (the underlined card is the off-suit):

With off-suit in low position:

A2345
23456
34567
45678
56789
6789T

There are six of them. If you try the same thing with the off-suit in the second lowest position you'll see that there are five of them:

23456
34567
45678
56789
6789T

The same is true for the off-suit in the third and fourth positions. For the off-suit in the highest position there are six of them since 6 through Jack is possible. Altogether, then, we have 27 Straights in this particular suit configuration. Multiplying this by 12 we end up with 324 Straights in all.

Well, I'm not going to drag you through all of these calculations. You can give counting them on your own a shot and check your results against the following chart. The numbers across the top of the chart represent the number of high cards, Jack through Ace, in the Flush part of the hand.

Hand

0

1

2

3

4

Total

RF4

0

0

0

624

156

780

Strt5

324

120

72

12

0

528

SF4

2,952

1,284

708

144

0

5,088

HiPr

0

3,924

5,064

1,116

0

10,104

RF3

0

0

5,808

3,720

0

9,528

FL4

16,380

47,088

22,044

0

0

85,512

Total

19,656

52,416

44,696

5,616

156

111,540

Breakdown of Four Flush Hands

I should point out that the breakdown in the above hands accounts for the hierarchy in hands when one uses optimum playing strategy.

Since one would not draw to a Flush on a five-card Straight or on a High Pair, we only consider the remaining hands. We would draw one card on the RF4, the SF4, and the FL4, a total of 91,380 hands. In this case of the 47 remaining cards in the deck nine are suited to the Flush. Thus the number of successful Flush draws will be 91,380 x 9/47 or 17,498.3. For the 9,528 RF3 hands we would draw 2. Here the correct multiplier is C(9, 2)/C(47, 2) or 0.0333 so we would end up with a four Flush in 9528 x 0.0333 or approximately 317.3 hands. Altogether, then, there are 17,498.3 + 317.3 or 17,815.6 hands in which a five Flush results from being dealt a four Flush.

Of the 5148 pat five Flushes, 160 of them contain a four-card Royal which takes precedence over the Flush. Four of these, however, contain a five-card Straight Flush of the form 9TJQK and this hand would be played as is. Hence there are 156 hands in which we would draw one card; in 4992 hands we would keep the pat Flush (including some Straight Flushes). Of the 47 cards remaining in the deck, 8 are suited to the Royal so we would end up with a Flush of some kind when drawing one card 156 x 8/47 or about 26.6 hands. Thus when dealt a pat Flush we would end up with a Flush 4992 + 26.6 or approximately 5018.6 times.

Here is the final tally. In the 2,598,960 possible starting Poker hands there are 17,815.6 + 5,018.6 or 22,834.2 final five Flushes that started out with either a pat Flush or a four Flush. This means that approximately 22% of these hands began with a pat five Flush. Naturally there are some Flushes that arise from drawing to other hands, but those other hands weren't relevant to analyzing my original concerns/perceptions. Even if we considered these other hands, it is clear that about one in five Flushes occur from drawing a pat Flush. Who'd a thunk it?

So when you're playing don't let your mind wander (or, for that matter, wonder). Concentrate on the game and just write to me to sort things out if things seem strange to you -- what you are seeing is probably just the way things are. See you next month.

Donald Catlin

Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers
Donald Catlin
Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers